Angles and the Unit Circle#

In this chapter we will define the unit circle and the various ways we make angular measurements on it.

Angle Measurements#

In cartesian space (an \(xy\)-grid) we define a ray along the \(x\) axis to be at zero degrees. We can rotate that ray counter-clockwise to an angle \(\phi\) and sweep around the \(xy\) plane.

_images/define-angle.svg

Fig. 2 Counterclockwise rotations are by definition positive (increasing in angle).#

A complete circle is defined to be \(360^\circ\). Manipulate the slider on the next plot to get an idea of where different angular measurements fall on the circle.

Arc Length and Radians#

The circumference of a circle is

(4)#\[C=2\pi r\]

We define a radian as the angle swept out by a line equal in length to the radius of the circle.

_images/what-is-a-radian.gif

Fig. 3 One radian is about 57.3 degrees.#

A radian is the “natural” unit for angles because it defines arcs around a circle in terms of its natural unit, \(\pi\). We can quickly convert between degrees and radians with

\[360^\circ=2\pi\,\text{radians}\]

Given an angle \(\phi\) measured in radians, we also define arc-length \(s\) as the length on the circumference of a circle.

(5)#\[s=r\phi\]

Arc-length as a beautifl result that helps us remember it by recognizing that we complete one complete loop, \(\phi=2\pi\), we will “travel” an arc-length equal to the circumference.

\[\text{Arc-length once around}: s = r(2\pi) = \text{the circumference}\]

Exercises#

  1. A car makes \(32^\circ\) turn on a road with a radius of curvature of 115 meters. How far did the car travel on that turn?

    Solution

    Don’t forget to convert the \(32^\circ\) to radians! Otherwise your answer will be in the wrong units.

    \[s=r\phi=(115\text{m})(32^\circ)\]
    \[=(115\text{m})\left(32^\circ\frac{2\pi}{360^\circ}\right)\]
    \[= 64.23 \text{m}\]

  2. How many times will 5.5 meter long rope wrap around an axle with a diameter of 8 centimeters?

    Solution

    Let \(L=5.5\) m and note that this will equal some number \(N\) multiple of the circumference.

    \[ L=N \cdot C \]
    \[ L=N \cdot 2\pi r \]

    Since we are given the diameter we’ll rewrite the previous equation using diameter, then solve for the number of turns \(N\).

    \[ L=N\cdot \pi D \]
    \[ N=\frac{L}{\pi D}\]
    \[ N=\frac{(5.5\,\text{m})}{\pi (0.08\,\text{m})} \]
    \[ N=21.8 \]